FedU and dFedU (Laplacian-Regularized Federated Multi-Task Learning)

A communication-centralized (FedU, Algorithm 1) and decentralized (dFedU, Algorithm 2) federated multi-task learning framework (Dinh, Vu, Tran, Dao, & Zhang 2022, Sections III-V, Equations 1-12, Algorithms 1-3, Theorems 1-2) that formulates client-relationship-aware learning over an undirected weighted entity graph G = (N, E, A) with graph Laplacian L = D - A (and Kronecker Laplacian L_cal = L otimes I_d) as min_{W = [w_1^T, ..., w_N^T]^T} { J(W) := sum_{k=1}^N F_k(w_k) + (eta / 2) sum_{k=1}^N sum_{l in N_k} a_{kl} ||w_k - w_l||^2 } (Equations 1-3): in each round t, active clients k in S^{(t)} perform R local stochastic gradient steps w_{k, r+1}^{(t)} = w_{k, r}^{(t)} - mu nabla F_tilde_k(w_{k, r}^{(t)}) toward their individual local optima, and the server (in FedU) or each client via direct neighbor exchange (in dFedU) applies the graph-Laplacian regularization update w_k^{(t+1)} = w_{k, R}^{(t)} - mu_tilde eta sum_{l in N_k} a_{kl} (w_{k, R}^{(t)} - w_{l, R}^{(t)}) with global step-size mu_tilde = mu R (Algorithms 1-3, Equation 45).